WASHINGTON — The Supreme Court on Monday debated whether the fees charged by mutual funds are too high, and if so, whether investors should be allowed to sue the boards that approved them.

The case put a spotlight on the often cozy relationship between mutual funds and their advisers. It is the advisers who sponsor the fund and choose members of its board, which in turn decides on fees that go to the advisers.

Lawyers for the Obama administration described these arrangements as “fraught with potential conflicts of interests,” and they urged the justices to open the door to suits against mutual funds if their fees are not in line with those charged to other, similar investors.

But they ran into sharply skeptical questions from Chief Justice John Roberts and Justice Antonin Scalia. They said investors can easily check the fees charged by a mutual fund and take their money elsewhere if the fees are too high.

“You can look it up on Morningstar,” Roberts said. “It’s right there. . . . It takes 30 seconds.”

Roberts and Scalia also questioned whether judges or juries should be given the duty of deciding which fees are too high. Say, for instance, a fund has a consistently above-average performance.

“What’s a fair fee in that situation?” Roberts asked.

Others, including Justice Sonia Sotomayor and Stephen Breyer, questioned whether the free market can be counted on to police these fees.

They took exception to the ruling last year from the 7th U.S. Circuit Court of Appeals, which tossed out a suit against Harris Associates of Chicago, which serves as the adviser for the Oakmark Funds. Several investors alleged Harris had charged the investors in the Oakmark funds fees that were allegedly twice as high as they charged to large pension funds.

The issue has drawn wide attention in the financial industry because the Supreme Court could set a new, stricter legal standard for mutual funds. About 91 million Americans, representing half of U.S. households, have money invested in mutual funds, the court was told.

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